Poincaré asymptotic series

Summary

A Poincaré asymptotic series of a function $f$ at infinity is a formal power series $\sum a_n z^{-n}$ with one property. Every partial sum approximates $f$ with an error smaller than the last power used. It is the asymptotic expansion for the asymptotic scale of the powers $z^{-n}$, as defined by Poincaré in 1886. The series may diverge for every $z$, and still give very accurate approximations. The coefficients are unique when they exist, but different functions can have the same series.